Infinite-order decay with derivatives

Let $q:\Omega\to(0,1]$ be a positive scale and $f$ smooth on $\Omega$ . Write $|f|_m(x)=\max_{|\alpha|\le m}|\partial^\alpha f(x)|$ , the size of its Cartesian jet. The function has infinite-order decay with derivatives as $q\to0$ if

\forall m,N\ge0\quad\exists C_{m,N},\delta_{m,N}>0: \quad |f|_m(x)\le C_{m,N}q(x)^N\quad\text{when }q(x)<\delta_{m,N}.

This is also called a flat remainder in the stated derivative topology.

Uniformity and extension

Constants are uniform over the specified points or labels, but may depend on $m,N$ . One positive decay order is insufficient. A bound for $f$ alone does not imply derivative bounds. When $q$ tends to zero at an ordinary smooth boundary and these estimates imply locally uniform vanishing of every mixed derivative there, the smooth zero-extension criterion applies. The definition itself concerns decay on $\Omega$ and does not assume that its missing boundary has already been parametrized.